Lateral Vibration of Bars. 135 



always be in excess of the true frequency. The method is 

 based on recognized dynamical principles, and when different 

 assumed types of vibration give different results for the 

 frequency, there is no room for doubt as to which is the 

 most nearly correct. 



The method advanced by Mr. Garrett, on the other hand, 

 is not apparently based on any dynamical principle. He 

 simply calculates the moment about the clamped end of what 

 he calls u the forces causing motion " and equates it to the 

 clastic couple at that end, both results being derived from an 

 assumed type of displacement which differs from that of the 

 actual dynamical problem. A mistake of sign really runs 

 through 3Ir. Garrett's work on pp. 583 & 584. His results, 

 e.g. his equation (5), require j/i/yi to be a positive quantity, 

 whereas it really equals — P, where k/2ir is the frequency. 

 To get the sign correct, Mr. Garrett ought to have employed 

 the •• reversed effective forces " instead of the " forces causing 

 motion. '' 



As to the types of vibration employed by Mr. Garrett in 

 his two illustrations, they are simply those employed by 

 Lord Rayleigh himself in his § 182. Mr. Garrett differs, 

 however, in the results he obtains for the frequency. Thus, 

 in the first case, &/2w denoting the frequency, and A a 

 certain function of the dimensions and physical properties of 

 the bar, the results obtained are as follows : — 



Garrett. Rayleigh. Exact theory. 



/j 2 = A(120/ll) A(140/ll) A(136-0/ll) 



The error in Garrett's result is thus almost exactly four 

 times that in Rayleigh's. The assumed type of vibration in 

 this first case answers to the deflexion of the bar by a statical 

 force at the end. In his next case Mr. Garrett, still 

 following Rayleigh, assumes the displacement to be of the 

 type answering to a statical force applied at some point 

 short of the end. The result (9) he obtains is again different 

 from Rayleigh's. In this instance the best result obtained 

 by Garrett (see his pp. 582 & 589) differs 1'4 per cent, from 

 the true frequency, whereas the best result obtained by 

 Rayleigh is only 0"2o per cent, in error. 



Mr. Garrett seems to see some difficulty in Rayleigh's 

 method which I fail to recognize. In such an elementary 

 case as a clamped-free bar it involves nothing more serious 

 than two simple integrals [e. g., in Garrett's first case 

 £*(3Z — x) 2 and (l — xf taken between and /], operations 

 which may surely be left to the reader of such a work as 

 Rayleigh's. 



